Abstract

The rotation method for the absolute testing of three flats is extended by adding a second rotation of one of the flats. This means that altogether five interferograms of pairs of flats (positional combinations) are evaluated: three basic combinations and two rotational combinations. The effect of random measuring errors is minimized by fully applying least-squares methods. Here the addition of a second rotation leads to a substantial increase of accuracy of the results and enables the lateral resolution to be further enhanced.

Fizeau arrangement. The surface distances dν are determined interferometrically, an additive constant is not of interest. Dν is the distance between the reference planes. The figure shows the combination AB in side view. The unknown flatness deviations xν and y−ν are to be determined. For each value of ν (Fig. 1) the azimuth νΦ/2 of flat A coincides with the azimuth −νΦ/2 of flat B. dν, Dν, xν, and y−ν, are functions of the position.

Top view of the four positional combinations of the rotation method used so far. In the fourth combination (ABΦ) flat B has been rotated from its position in the first combination (AB) by the angle Φ. In the first three combinations (the basic combinations) the azimuths ϑ = 0 of both flats coincide (coincidence at ν = 0); in the fourth combination (rotational combination) the azimuths ϑ = Φ/2 coincide (coincidence at ν = 1).

Survey of the error propagation factors fν(x)(N) of flat A if one set of four positional combinations according to Fig. 3 is evaluated. fmin(x)(N) is the minimum and fmax(x)(N) is the maximum of the 2N values fν(x)(N) (ν = −N + 1, −N + 2, … N), whereas fmean(x)(N) is their quadratic mean:
fmean(x)(N)={∑ν=-N+1N[fν(x)(N)]2/2N}1/2.The error propagation factors describe the transition of the mean measuring error to the mean error of the results.

Top view of the five positional combinations of the extended rotation method. The first four combinations are the same as in Fig. 3 with Φ = 360°/N, whereas in the fifth combination (ABKΦ) flat B has been rotated from its position in the first combination (AB) by the angle KΦ (K = 3 in this example). In ABKΦ the azimuths ϑ = KΦ/2 coincide (coincidence at ν = K).

Examples of the quadratic mean of the error propagation factors according to Eq. (9) as a function of K with the parameter N; N and K correspond to Eqs. (6) and (7). Note the minimum of fmean(x)(N, K) for
K≈N;N = 16, 20, 25, 30, 36.

Survey of the error propagation factors
fν(x)(N,N) of flat A if one set of five positional combinations according to Fig. 5 is evaluated. N = 4, 9, 16, … 100.
fmin(x)(N,N) is the minimum and
fmax(x)(N,N) is the maximum of the 2N values
fν(x)(N,N) (ν = −N + 1, −N + 2, …N);
fmean(x)(N,N) is their quadratic mean according to Eq. (9) with
K=N; cf. Fig. 4.

Example of a combined enhancement of resolution in depth and of lateral resolution if, instead of four positional combinations with N = 20, five combinations with N = 100 are used. Four positional combinations according to Section 1; five positional combinations according to Section 2.

Reduction of the density of information near the center of each of the three surfaces to be tested. For ρ > ρ0 all five combinations are evaluated, whereas for ρ ≤ ρ0 only the four combinations AB, BC, CA, and ABKΦ are evaluated. The figure shows the example N = 36, K = 4.